A Beginners’ Guide to Modelling of Electric Double Layer under Equilibrium, Nonequilibrium and AC Conditions

doi:10.13208/j.electrochem.210847

Abstract

Abstract:

In electrochemistry, perhaps also in other time-honored scientific disciplines, knowledge labelled classical usually attracts less attention from beginners, especially those pressured or tempted to quickly jam into research fronts that are labelled, not always aptly, modern. In fact, it is a normal reaction to the burden of history and the stress of today. Against this context, accessible tutorials on classical knowledge are useful, should some realize that taking a step back could be the best way forward. This is the driving force of this article themed at physicochemical modelling of the electric (electrochemical) double layer (EDL). We begin the exposition with a rudimentary introduction to key concepts of the EDL, followed by a brief introduction to its history. We then elucidate how to model the EDL under equilibrium, using firstly the orthodox Gouy-Chapman-Stern model, then the symmetric Bikerman model, and finally the asymmetric Bikerman model. Afterwards, we exemplify how to derive a set of equations governing the EDL dynamics under nonequilibrium conditions using a unifying grand-potential approach. In the end, we expound on the definition and mathematical foundation of electrochemical impedance spectroscopy (EIS), and present a detailed derivation of an EIS model for a simple EDL. We try to avoid the omission of supposedly ‘trivial’ information in the derivation of models, hoping that it can ease the access to the wonderful garden of physical electrochemistry.

Key words: electric double layer, equilibrium, nonequilibrium, electrochemical impedance spectroscopy

Lu-Lu Zhang, Chen-Kun Li, Jun Huang. A Beginners’ Guide to Modelling of Electric Double Layer under Equilibrium, Nonequilibrium and AC Conditions[J]. Journal of Electrochemistry, 2022, 28(2): 2108471.

Figures/Tables 13

References 60

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Symbol (unit)	Value	Physical significance	Note
Constants
k_B(J·K^-1)	1.381 × 10^-23	Boltzmann constant
T(K)	298.15	Absolute temperature
h(J·s)	6.626 × 10^-34	Planck constant
e₀(C)	1.602 × 10^-19	Elementary charge
N_A(mol^-1)	6.022 × 10²³	Avogadro constant
$\epsilon_{0}$(F·m^-1)	8.854 × 10^-12	Vacuum permittivity
F(C·mol^-1)	96485	Faraday constant
R(J·K^-1·mol^-1)	8.314	Gas constant
Solution properties
$\epsilon_{HP}$(F·m^-1)	6$\epsilon_{0}$	Dielectric permittivity of the space between the electrode and the HP	Ref [56]
$\epsilon_{S}$(F·m^-1)	78.5$\epsilon_{0}$	Bulk dielectric permittivity of the water solvent medium
δ_HP(nm)	0.4125	Distance from the electrode to the HP, calculated by 1.5 d with the diameter of water d = 0.275 nm.
D(m²·s^-1)	1 × 10^-9	Diffusion coefficient
c^b(mol·m^-3)	1	Concentration of total cations (anions) in the bulk solution
c_A⁰,c_B⁰,c_H+⁰(mol·m^-3)	1	Concentrations of B, A and H⁺ under standard conditions
Electrode properties
ϕ_pzc(V_SHE)	0.3	Potential of zero charge	Estimated
a_M(Å)	3.5	Lattice constant of the electrode	Estimated
n_M(m^-2)	4.713 × 10¹⁸	Areal number density of M(111), calculated by ($\sqrt{3}$a_M²)^-1
Reaction properties
E⁰⁰(V_SHE)	0.6	Equilibrium potential of the reaction at standard state	Estimated
ΔG_a⁰⁰(eV)	0.4	Activation energy of the reaction at standard equilibrium state	Estimated